The way to become a member of permanent staff for the External Service is to get through an open competition exam.
However, Delegations make extensive use of local agents who are generally employed on an indefinite basis, even though the contractual conditions differ from those of officials. Vacancies for local agents in Delegations are generally published in the most popular newspapers in the country of interest.
As regards temporary jobs, you may find of interest:
Find out more about a career as a member of the European Commission External Service.